nLab
little site

Context

Topos Theory

Could not include topos theory - contents

Little sites

Idea

Let CC be a category with a pretopology JJ (i.e. a site) and aa an object of CC. As an analogy with sheaves on a topological space XX, which are defined on the site Op(X)Op(X) of open sets of XX, we can try to define sheaves on aa, using the elements of covering families of aa from JJ. This is called the little site of aa, in contrast to the big site of aa which is the slice category C/aC/a with its induced topology.

The topos of sheaves on the little site is the petit topos of aa.

A little site may sometimes be called a small site, but it's probably best to save that name for a site which is a small category.

Definition

David Roberts: The following is experimental, use at own risk, although I’m sure it has been thought about before.

Consider the subcategory J/aJ/a of C/aC/a with objects u 0au_0 \to a such that u 0u_0 is a member of some covering family U={u ia}U = \{u_i \to a\}. Given two such objects u 0au_0 \to a, v 0av_0 \to a, and covering families UU, VV that contain them, there is a covering family W=UVW = UV which is the pullback (or at least a weak pullback) of UU and VV in CC. There is then some element ww of WW such that there is a square

w v 0 u 0 a\array{ w & \to & v_0 \\ \downarrow & & \downarrow \\ u_0 &\to & a }

so J/aJ/a is ‘a bit like’ the category of opens of a space (it’s probably cofiltered, but I haven’t checked that there are weak equalisers).

Now the morphisms of J/aJ/a are those triangles

v 0 u 0 a.\array{ v_0 & \to & u_0 \\ & \searrow& \downarrow \\ & & a }\, .

such that v 0u 0v_0 \to u_0 is an element of a covering family of u 0u_0, so the arrows wu 0w \to u_0 and wv 0w \to v_0 really are morphisms of J/aJ/a. Then we say a covering family of u 0au_0\to a is a collection of triangles that, when we forget the maps to aa, form a covering family of u 0u_0 in CC. This is at the very least a coverage, and so we have a site.

To be continued…

Revised on November 16, 2010 18:15:54 by Urs Schreiber (131.211.232.149)